A beautiful landscape picture, a beautiful piano-sonata that leaves words that describe aesthetics and beauty when it comes to art and music. What about mathematics, can we talk about an abstract idea with the word "beautiful", in addition to commonly used words like useful, intelligent?
Answer is possible. And it's not just the judgment of mathematicians, but ordinary people also agree. That is the conclusion from a study published in the international journal Cognition Science published this month (August 2019).
Accordingly, ordinary Americans (not trained in math) also evaluate mathematical formulas about their beauty in the same way they value art or music. The beauty of mathematics that they realize is not only beauty in one direction. 300 participants were asked to evaluate the specific beauty in four different mathematical formulas according to 9 aesthetic criteria such as elegance, complexity, universality .
The question of mathematical aesthetics began when Stefan Steinerberger, Yale's assistant professor of mathematics, co-authored, compared a proof he was teaching with "a very good Schubert sonata". The students studying mathematics at Yale also had an impressive amount of knowledge about music, so after three or four people went to see Stefan and asked him what it meant. After an interview with psychology, Stefan was introduced to Samuel GB Johnson, co-author of research, assistant professor of marketing at the University of Bath. Samuel is working on a doctorate in psychology at Yale, studying arguments and decision making, how we humans evaluate different explanations and formulas for all things.
After that, the two unified the way to design the experiment to test Stefan's hypothesis: whether we share the sensitivity to beauty of math as other fields like art or music, and beyond this is true for mathematicians.
The study uses 4 mathematical formulas (proofs), 4 landscape paintings and 4 piano sonatas. The similarity between music and mathematics has long been noted, so the authors wanted to test another aesthetic field, the art, to see how we value aesthetics with something more universal.
The study consists of 3 parts: In the first part, a group is required to match 4 math formulas with 4 landscape paintings based on the aesthetic similarity they recognize. In part 2, another group matched the math formulas with sonatas, and part 3 was another group that independently voted each of the 4 artworks and 4 math formulas on 9 different criteria and Total points for beauty follow the scale from 0 to 10.
The evaluation criteria are taken from the essay on mathematical beauty "A Mathematician's Apology" by GH Hardy. GH Hardy, full name Godfrey Harold Hardy is a very famous British mathematician in the field of calculus and number theory with a typical work of the Hardy-Littlewood hypothesis. Hardy's 6 criteria have been expanded into 9 criteria: seriousness, universality, profoundness, novelty, clarity, simplicity, elegance, complexity and sophistication.
4 mathematical formulas used in the study are quite basic, easy to understand. Here is one of them.
A formula for calculating the sum of a geometric sequence is used to evaluate mathematical beauty
Threads require the sum of infinite numbers 1/2 + 1/4 + 1/8 + 1/16 + ··· by geometry. We can prove this by cutting the square in the upper image into pieces with a total area equal to 1.
The voting results of the participating group 3 show that artistic works and mathematical formulas with high marks of elegance will predict a high score on beauty. After examining the similarities in the evaluation scores of group 3 and group 1 (which requires more simple work than matching similar math formulas and art works with similar aesthetics), the authors surprise because the similarity of group 3 can predict the results of group 1. The participants of group 3 have the consensus to give points to the formulas and pictures of elegance, and group participants 1 also has a similar tendency: formulas and pictures that are considered to be the most elegant by the third group are all connected in the results of group 1.
Thus, ordinary people who do not study mathematics do not only have the same recognition of mathematical and artistic beauty but also recognize the similarity between mathematics and art. In other words, they also agree on how to evaluate the elements that make up beauty, regardless of what beauty is in the field.
Stefan hopes the research will be done again with other musical works, proofs and art."We have proven this phenomenon but we do not know its limits. It will not exist anywhere? Is it necessary to be classical music? Needed pictures of the natural world inherent High aesthetics? ".
Both authors said that this study has a lot of meaning in math education, especially at the secondary level. That will create "opportunities for making mathematical aspects more abstract and more accessible and more enjoyable for high school students" , and "can also be useful in the perspective of promotion." encourage more people to enter mathematics " .
"Understanding what people think is beautiful in mathematics will help you understand how people understand math at the beginning and how they handle it . " And there are also hidden human understandings of this question, that's how we really think about things as people. To understand this, we have "obligations" to cooperate with psychologists, Stefan added.